The life in hours of a battery is known to be approximately...

The life in hours of a battery is known to be approximately normally distributed with standard deviation $\sigma=1.25$ hours. A random sample of 10 batteries has a mean life of $\bar{x}=$ 40.5 hours. a. Is there evidence to support the claim that battery life exceeds 40 hours? Use $\alpha=0.5$. b. What is the $P$ -value for the test in part (a)? c. What is the $\beta$ -error for the test in part (a) if the true mean life is 42 hours? d. What sample size would be required to ensure that $\beta$ does not exceed 0.10 if the true mean life is 44 hours? e. Explain how you could answer the question in part (a) by calculating an appropriate confidence bound on battery life

The life in hours of a battery is known to be approximately normally distributed with standard deviation $\sigma=1.25$ hours. A random sample of 10 batteries has a mean life of $\bar{x}=$ 40.5 hours.
a. Is there evidence to support the claim that battery life exceeds 40 hours? Use $\alpha=0.5$.
b. What is the $P$ -value for the test in part (a)?
c. What is the $\beta$ -error for the test in part (a) if the true mean life is 42 hours?
d. What sample size would be required to ensure that $\beta$ does not exceed 0.10 if the true mean life is 44 hours?
e. Explain how you could answer the question in part (a) by calculating an appropriate confidence bound on battery life

Key Concepts

Sample Size Determination for Desired Power

This concept deals with calculating the number of observations required to achieve a specific statistical power (or equivalently, a bound on ?-error) for detecting an effect of a given size. It involves understanding the relationships between the effect size, variability, significance level, and the power requirements, ensuring that the study is adequately powered to detect meaningful differences.

Confidence Intervals/Bounds in Hypothesis Testing

Confidence intervals provide a range of plausible values for a population parameter based on sample data. In the context of hypothesis testing, if a hypothesized value lies outside a 100(1??)% confidence interval, it implies that there is sufficient evidence to reject the null hypothesis. This approach can be used as an alternative to directly computing a test statistic and p-value, offering another perspective on the evidence against the null hypothesis.

Type II Error (?-error) and Statistical Power

Type II error, denoted as ?, is the probability of failing to reject the null hypothesis when it is actually false. The complement of this error (1??) is known as the power of the test, which is the likelihood that the test will correctly detect an effect when there is one. Calculating ? helps in understanding the risk of missing a true effect, and it is crucial in planning studies and determining sample sizes.

Normal Distribution

This concept refers to the continuous probability distribution that is symmetric about the mean. It is characterized by its mean and standard deviation, and many statistical methods assume that data are normally distributed. In hypothesis testing problems like this one, the normal distribution is used to model the distribution of sample means, especially when the population standard deviation is known.

One-Sided (One-Tailed) Test

A one-sided test is used when the alternative hypothesis specifies a direction of the effect—in this case, that the battery life exceeds a particular value. This test only considers the probabilities in one tail of the distribution, which affects the critical values and the power of the test.

Hypothesis Testing for a Mean with Known ?

When the population standard deviation is known, the z-test is employed to test hypotheses about the population mean. This involves comparing a test statistic, calculated from the sample data, to critical values derived from the standard normal distribution. It is used to determine whether there is sufficient evidence to reject a stated null hypothesis in favor of an alternative hypothesis.

P-value

The p-value is the probability of obtaining a test statistic as extreme as or more extreme than the one observed, assuming that the null hypothesis is true. It quantifies the strength of evidence against the null hypothesis, with a smaller p-value indicating stronger evidence to support the alternative hypothesis.

Transcript

00:01 Okay, the life of a battery is known to be approximately normally distributed with the standard deviation of 1 .25 hours.

00:10 A random sample of 10 batteries has a mean life of 40 .5 hours.

00:16 Is there evidence, question a, to support the claim that battery life exceeds 40 hours? and we are told to use a 0 .5 level of significance.

00:36 Okay, in our case, we need to test whether or better, life exceeds 40 hours or not.

00:41 So we're going to take the null hypothesis as this equals 40.

00:53 The alternative pulse hypothesis will be is greater than 40.

01:10 Let's recall the rejection, recall that the rejection criteria for a fixed level test states that for an alternative hypothesis h1 of form u is greater than the null hypothesis the null hypothesis is rejected okay so let's um do this okay so let's compute compute a test statistic and this will be equal to 40 .5 minus 40 over 125 divided by the square root of 10 and this will equal about um 1 .2 now let's compute, this was z0, let's compute z using the normal table for this equals 0 .5 for the given equation.

03:32 And we get, okay, now note that the value of the test statistics, so z is greater than this value.

03:47 So we get the battery life, the rejection criteria and the null hypothesis is rejected, and the battery life is greater than 40 hours.

04:32 Okay, we have several questions to do here.

04:38 What is p value for part a? okay, so p equal one minus p times c zero.

05:20 So we're going to get, so this was equal to one, the value of the test statistic.

05:48 Hang on just one second here, 1 .265.

05:56 So we've got 1 .265.

06:04 So we're going to have 1 minus p times z is less than or equal to 1 .265.

06:21 And this will be 1 minus 0 .896 -165, 0 .103835.

06:47 And i hope that this is the value that i need.

06:49 I hope like, heck, that's the value i need.

06:54 Okay, let's do c.

07:01 What is the beta error? if the true mean life is 42 hours.

07:24 Okay.

07:31 So our beta is going to equal where this is true for standard normal distribution, z and delta equals x minus standard deviation.

08:25 Okay, so let's do this.

08:39 So we're going to have our beta will equal two times the square root of 10 over 1 .25.

08:56 And that'll be approximately equal to 0 .00 -0313...

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